Polytopes and Machine Learning

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.

A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends severa...

We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from different fields ranging from geometry to representation theory, from combinatorics to number theory, we present a comparative study of the accuracies on different problems. The paradigm should be useful for conjecture formulation, finding more eff...

We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large majority of the projections lie on vertices of the polytopes. To do these projections efficiently we derive a vertex-oriented incremental algorithm to project a point onto any arbitrary polytope, as well as give specific algorithms to cater to sim...

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological appli...

The problem of classifying high-dimensional shapes in real-world data grows in complexity as the dimension of the space increases. For the case of identifying convex shapes of different geometries, a new classification framework has recently been proposed in which the intersections of a set of one-dimensional representations, called rays, with the boundaries of the shape are used to identify the specific geometry. This ray-based classification (RBC) has been empirically verif...

The Separating Hyperplane theorem is a fundamental result in Convex Geometry with myriad applications. Our first result, Random Separating Hyperplane Theorem (RSH), is a strengthening of this for polytopes. $\rsh$ asserts that if the distance between $a$ and a polytope $K$ with $k$ vertices and unit diameter in $\Re^d$ is at least $\delta$, where $\delta$ is a fixed constant in $(0,1)$, then a randomly chosen hyperplane separates $a$ and $K$ with probability at least $1/poly(...

Voronoi tessellations are used to partition the Euclidean space into polyhedral regions, which are called Voronoi cells. Labeling the Voronoi cells with the class information, we can map any classification problem into a Voronoi tessellation. In this way, the classification problem changes into a query of just finding the enclosing Voronoi cell. In order to accomplish this task, we have developed a new algorithm which generates a labeled Voronoi tessellation that partitions t...

This is the first chapter in our "Toric Topology" book project. Further chapters are coming. Comments and suggestions are very welcome.

We make a connection between classical polytopes called zonotopes and Support Vector Machine (SVM) classifiers. We combine this connection with the ellipsoid method to give some new theoretical results on training SVMs. We also describe some special properties of soft margin C-SVMs as parameter C goes to infinity.